Video Transcript
A spacecraft heading toward the
Moon to deploy a rover on the surface has a mass of 870 kilograms. The magnitude of the gravitational
potential energy of the spacecraft-and-Moon system is 427 megajoules. How far away from the Moon’s center
of mass is the spacecraft? Use a value of 7.35 times 10 to the
22nd kilograms for the mass of the Moon and 6.67 times 10 to the negative 11th cubic
meters per kilogram second squared for the universal gravitational constant. Give your answer to three
significant figures.
If we say that this pink circle is
the Moon and that this is our spacecraft heading toward it, our problem statement
tells us the mass of the spacecraft, let’s call that 𝑚 sub s; the mass of the moon,
we’ll call that 𝑚 sub m. And it also tells us the
gravitational potential energy of this system of two masses, the spacecraft and the
Moon. We’ll call that magnitude
𝐺𝑃𝐸. Knowing all this, what we want to
solve for is the distance between the spacecraft and the center of the Moon. And we can call that distance
𝑟.
To get started, we can recall the
equation for gravitational potential energy in a radial gravitational field. This is the kind of field created
by a spherically symmetric object, such as the Moon. For this kind of field, the
gravitational potential energy shared between two masses, the larger one creating
the field and the smaller one experiencing it, is equal to negative 𝐺, the
universal gravitational constant, times those masses divided by the distance between
their centers. In our scenario, it’s not 𝐺𝑃𝐸 we
want to solve for though, but rather the distance 𝑟.
If we multiply both sides of this
equation by the fraction 𝑟 divided by 𝐺𝑃𝐸, then on the left-hand side, that
gravitational potential energy cancels out. And on the right, the distance 𝑟
cancels. And that leaves us with this
equation. 𝑟 is equal to negative big 𝐺
times big 𝑀 times little 𝑚 divided by 𝐺𝑃𝐸.
One important thing to remember
here is that this term, the gravitational potential energy in this equation, is
negative. And so if we were to plug in actual
values in this equation, we would have a negative multiplied by a negative, which is
a positive, giving us then an overall positive value for 𝑟. We can apply this equation to our
scenario like this. The distance between the center of
the Moon and the spacecraft is equal to the universal gravitational constant times
the mass of the Moon times the mass of the spacecraft all divided by the magnitude
of the gravitational potential energy of this two-mass system. And notice that we’re given all of
the values that appear on the right-hand side of this equation.
Our next step then is to substitute
them in. So here we have the value for the
universal gravitational constant. Here’s the mass of the Moon, here’s
the mass of our spacecraft, and here’s the magnitude of the gravitational potential
energy between these masses. As a last step, before we calculate
𝑟, we’ll want to convert this gravitational potential energy from units of
megajoules into units of joules. That way, all the units in this
expression will be on the same footing. Now, one megajoule is equal to 10
to the sixth or a million joules. So we can rewrite our denominator
as 427 times 10 to the sixth joules. This number isn’t in scientific
notation, but it is the correct number of joules of energy. When we calculate 𝑟, to three
significant figures, we get a result of 9.99 times 10 to the sixth meters. We can equivalently express this as
9,990 kilometers. That’s the distance from the center
of mass of the Moon to the spacecraft.
